Old bathtub drain problem

I live in an older building, built around 1930, with a remodel that was done probably around 40 years ago. My problem is this, my bathtub drain has stopped draining quickly, it gets the water gone, but takes its sweet time getting there. I have tried most of the natural remedies (baking soda, plunger, boiling water) and can make it a little bit better, but only for about a month or so.

I bought a small affordable snake, but there seems to be a grating blocking my way getting it down the bathtub drain about an inch down.

I rent, and can't do anything drastic, and as long as the drain is draining, the landlord won't do much.